15-462 Computer Graphics I Lecture 17 Spatial Data Structures Hierarchical Bounding Volumes Regular Grids Octrees BSP Trees Constructive Solid Geometry (CSG) April 1, 2003 [Angel 9.10] Frank Pfenning Carnegie Mellon University http://www.cs.cmu.edu/~fp/courses/graphics/
Ray Tracing Acceleration Faster intersections Faster ray-object intersections Object bounding volume Efficient intersectors Fewer ray-object intersections Hierarchical bounding volumes (boxes, spheres) Spatial data structures Directional techniques Fewer rays Adaptive tree-depth control Stochastic sampling Generalized rays (beams, cones) 04/01/2003 15-462 Graphics I 2
Spatial Data Structures Data structures to store geometric information Sample applications Collision detection Location queries Chemical simulations Rendering Spatial data structures for ray tracing Object-centric data structures (bounding volumes) Space subdivision (grids, octrees, BSP trees) Speed-up of 10x, 100x, or more 04/01/2003 15-462 Graphics I 3
Bounding Volumes Wrap complex objects in simple ones Does ray intersect bounding box? No: does not intersect enclosed objects Yes: calculate intersection with enclosed objects Common types Boxes, axis-aligned Boxes, oriented Spheres Finite intersections or unions of above 04/01/2003 15-462 Graphics I 4
Selection of Bounding Volumes Effectiveness depends on: Probability that ray hits bounding volume, but not enclosed objects (tight fit is better) Expense to calculate intersections with bounding volume and enclosed objects Amortize calculation of bounding volumes Use heuristics good bad 04/01/2003 15-462 Graphics I 5
Hierarchical Bounding Volumes With simple bounding volumes, ray casting still has requires O(n) intersection tests Idea use tree data structure Larger bounding volumes contain smaller ones etc. Sometimes naturally available (e.g. human figure) Sometimes difficult to compute Often reduces complexity to O(log(n)) 04/01/2003 15-462 Graphics I 6
Ray Intersection Algorithm Recursively descend tree If ray misses bounding volume, no intersection If ray intersects bounding volume, recurse with enclosed volumes and objects Maintain near and far bounds to prune further Overall effectiveness depends on model and constructed hierarchy 04/01/2003 15-462 Graphics I 7
Spatial Subdivision Bounding volumes enclose objects, recursively Alternatively, divide space For each segment of space keep list of intersecting surfaces or objects Basic techniques Regular grids Octrees (axis-aligned, non-uniform partition) BSP trees (recursive Binary Space Partition, planes) 04/01/2003 15-462 Graphics I 8
Grids 3D array of cells (voxels) that tile space Each cell points to all intersecting surfaces Intersection alg steps from cell to cell 04/01/2003 15-462 Graphics I 9
Caching Intersection points Objects can span multiple cells For A need to test intersection only once For B need to cache intersection and check next cell for closer one If not, C could be missed (yellow ray) A B C 04/01/2003 15-462 Graphics I 10
Assessment of Grids Poor choice when world is non-homogeneous Size of grid Too small: too many surfaces per cell Too large: too many empty cells to traverse Can use alg like Bresenham s for efficient traversal Non-uniform spatial subdivision more flexible Can adjust to objects that are present 04/01/2003 15-462 Graphics I 11
Outline Hierarchical Bounding Volumes Regular Grids Octrees BSP Trees Constructive Solid Geometry (CSG) 04/01/2003 15-462 Graphics I 12
Quadtrees Generalization of binary trees in 2D Node (cell) is a square Recursively split into 4 equal sub-squares Stop subdivision based on number of objects Ray intersection has to traverse quadtree More difficult to step to next cell 04/01/2003 15-462 Graphics I 13
Octrees Generalization of quadtree in 3D Each cell may be split into 8 equal sub-cells Internal nodes store pointers to children Leaf nodes store list of surfaces Adapts well to non-homogeneous scenes 04/01/2003 15-462 Graphics I 14
Assessment for Ray Tracing Grids Easy to implement Require a lot of memory Poor results for non-homogeneous scense Octrees Better on most scenes (more adaptive) Alternative: nested grids Spatial subdivision expensive for animations Hierarchical bounding volumes Natural for hierarchical objects Better for dynamic scenes 04/01/2003 15-462 Graphics I 15
Other Spatial Subdivision Techniques Relax rules for quadtrees and octrees k-dimensional tree (k-d tree) Split at arbitrary interior point Split one dimension at a time Binary space partitioning tree (BSP tree) In 2 dimensions, split with any line In k dims. split with k-1 dimensional hyperplane Particularly useful for painter s algorithm Can also be used for ray tracing [see handout] 04/01/2003 15-462 Graphics I 16
Outline Hierarchical Bounding Volumes Regular Grids Octrees BSP Trees Constructive Solid Geometry (CSG) 04/01/2003 15-462 Graphics I 17
BSP Trees Split space with any line (2D) or plane (3D) Applications Painters algorithm for hidden surface removal Ray casting Inherent spatial ordering given viewpoint Left subtree: in front, right subtree: behind Problem: finding good space partitions Proper ordering for any viewpoint Balance tree For details, see http://reality.sgi.com/bspfaq/ 04/01/2003 15-462 Graphics I 18
Building a BSP Tree Use hidden surface removal as intuition Using line 1 or line 2 as root is easy Line 1 3 2 1 1 A D C Line 3 Line 2 B A C D a BSP tree using 2 as root 3 B the subdivision of space it implies 2 Viewpoint 04/01/2003 15-462 Graphics I 19
Splitting of surfaces Using line 3 as root requires splitting Line 1 3 Line 2a 2b 2a Line 3 Line 2b 1 Viewpoint 04/01/2003 15-462 Graphics I 20
Building a Good Tree Naive partitioning of n polygons yields O(n 3 ) polygons (in 3D) Algorithms with O(n 2 ) increase exist Try all, use polygon with fewest splits Do not need to split exactly along polygon planes Should balance tree More splits allow easier balancing Rebalancing? 04/01/2003 15-462 Graphics I 21
Painter s Algorithm with BSP Trees Building the tree May need to split some polygons Slow, but done only once Traverse back-to-front or front-to-back Order is viewer-direction dependent What is front and what is back of each line changes Determine order on the fly 04/01/2003 15-462 Graphics I 22
Details of Painter s Algorithm Each face has form Ax + By + Cz + D Plug in coordinates and determine Positive: front side Zero: on plane Negative: back side Back-to-front: inorder traversal, farther child first Front-to-back: inorder traversal, near child first Do backface culling with same sign test Clip against visible portion of space (portals) 04/01/2003 15-462 Graphics I 23
Clipping With Spatial Data Structures Accelerate clipping Goal: accept or rejects whole sets of objects Can use an spatial data structures Scene should be mostly fixed Terrain fly-through Gaming Hierarchical bounding volumes Octrees 04/01/2003 15-462 Graphics I 24
Data Structure Demos BSP Tree construction http://symbolcraft.com/graphics/bsp/index.html KD Tree construction http://www.rolemaker.dk/nonrolemaker/uni/algogem/kdtree.htm 04/01/2003 15-462 Graphics I 25
Real-Time and Interactive Ray Tracing Interactive ray tracing via space subdivision http://www.cs.utah.edu/~reinhard/egwr/ Interactive ray tracing with good hardware http://www.cs.utah.edu/vissim/projects/raytracing/ 04/01/2003 15-462 Graphics I 26
Outline Hierarchical Bounding Volumes Regular Grids Octrees BSP Trees Constructive Solid Geometry (CSG) 04/01/2003 15-462 Graphics I 27
Constructive Solid Geometry (CSG) Generate complex shapes with simple building blocks (boxes, spheres, cylinders, cones,...) Particularly applicable for machined objects Efficient with ray tracing 04/01/2003 15-462 Graphics I 28
Example: A CSG Train Brian Wyvill et al., U. of Calgary 04/01/2003 15-462 Graphics I 29
Boolean Operations Intersection and union Subtraction Example: drilling a hole Subtract From To get 04/01/2003 15-462 Graphics I 30
CSG Trees Set operations yield tree-based representation Use these trees for ray/objects intersections Think about how! 04/01/2003 15-462 Graphics I 31
Implicit Functions for Booleans Solid as implicit function, F(x,y,z) F(x, y, z) < 0 interior F(x, y, z) = 0 surface F(x, y, z) > 0 exterior For CSG, use F(x, y, z) 2 {-1, 0, 1} F A Å B (p) = max (F A (p), F B (p)) F A [ B (p) = min (F A (p), F B (p)) F A B (p) = max (F A (p), - F B (p)) 04/01/2003 15-462 Graphics I 32
Summary Hierarchical Bounding Volumes Regular Grids Octrees BSP Trees Constructive Solid Geometry (CSG) 04/01/2003 15-462 Graphics I 33
Preview Radiosity Assignment 6 due Thursday Assignment 7 (ray tracing) out Thursday 04/01/2003 15-462 Graphics I 34